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Theorem ofsubeq0 9759
Description: Function analog of subeq0 9089. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubeq0  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )

Proof of Theorem ofsubeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5405 . . . . . . 7  |-  ( F : A --> CC  ->  F  Fn  A )
31, 2syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
4 simp3 957 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5405 . . . . . . 7  |-  ( G : A --> CC  ->  G  Fn  A )
64, 5syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
7 simp1 955 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3391 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2297 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2297 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6103 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  o F  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
12 c0ex 8848 . . . . . . 7  |-  0  e.  _V
1312fvconst2 5745 . . . . . 6  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1413adantl 452 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
1511, 14eqeq12d 2310 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  =  0 ) )
16 ffvelrn 5679 . . . . . 6  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
171, 16sylan 457 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
18 ffvelrn 5679 . . . . . 6  |-  ( ( G : A --> CC  /\  x  e.  A )  ->  ( G `  x
)  e.  CC )
194, 18sylan 457 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
20 subeq0 9089 . . . . 5  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC )  -> 
( ( ( F `
 x )  -  ( G `  x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
2117, 19, 20syl2anc 642 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F `  x
)  -  ( G `
 x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
2215, 21bitrd 244 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
2322ralbidva 2572 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A. x  e.  A  ( ( F  o F  -  G
) `  x )  =  ( ( A  X.  { 0 } ) `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
243, 6, 7, 7, 8offn 6105 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  -  G )  Fn  A )
2512fconst 5443 . . . 4  |-  ( A  X.  { 0 } ) : A --> { 0 }
26 ffn 5405 . . . 4  |-  ( ( A  X.  { 0 } ) : A --> { 0 }  ->  ( A  X.  { 0 } )  Fn  A
)
2725, 26ax-mp 8 . . 3  |-  ( A  X.  { 0 } )  Fn  A
28 eqfnfv 5638 . . 3  |-  ( ( ( F  o F  -  G )  Fn  A  /\  ( A  X.  { 0 } )  Fn  A )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
2924, 27, 28sylancl 643 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
30 eqfnfv 5638 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
313, 6, 30syl2anc 642 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
3223, 29, 313bitr4d 276 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    - cmin 9053
This theorem is referenced by:  psrridm  16165  dv11cn  19364  coeeulem  19622  plydiveu  19694  facth  19702  quotcan  19705  plyexmo  19709  mpaaeu  27458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055
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